We show that if *C* is a supersingular genus-2 curve over an
algebraically-closed field of characteristic 2, then there are infinitely
many Richelot isogenies starting from *C*. This is in contrast to what
happens with non-supersingular curves in characteristic 2, or to arbitrary
curves in characteristic not 2: In these situations, there are at most fifteen
Richelot isogenies starting from a given genus-2 curve.

More specifically, we show that if *C*_{1} and *C*_{2}
are two arbitrary supersingular genus-2 curves over an algebraically-closed
field of characteristic 2, then there are exactly sixty Richelot isogenies from
*C*_{1} to *C*_{2}, unless either *C*_{1}
or *C*_{2} is isomorphic to the curve
*y*^{2} + *y* = *x*^{5}. In that case, there are
either twelve or four Richelot isogenies from *C*_{1} to
*C*_{2}, depending on whether *C*_{1} is isomorphic to
*C*_{2}. (Here we count Richelot isogenies up to isomorphism.) We
give explicit constructions that produce all of the Richelot isogenies between
two supersingular curves.